Binomial Expansion

( Positive Integer Powers )

Factorial Notation

When you see an exclamation mark following a number in mathematics it is known as a factorial. For example, 6! is said ‘6 factorial’ and you multiply all of the positive integers less than 6 together:

$6!=6\times 5\times 4 \times 3 \times 2 \times 1=720$

Here are some more examples:

$8!=8\times 7\times 6\times 5\times 4 \times 3 \times 2 \times 1=40,320$

$4!\times 3!=4\times 3\times 2 \times 1\times 3\times 2 \times 1=24\times 6=144$

$9!\div 5!=\frac{9\times 8\times 7\times 6\times 5\times 4 \times 3 \times 2 \times 1}{5\times 4 \times 3 \times 2 \times 1}=9\times 8\times 7\times 6=3,024$

Pascal’s Triangle and ‘n choose r’

Pascal’s triangle is the pyramid of numbers where each number is formed by adding together the two numbers that are directly above it:

The triangle continues on this way and is named after a French mathematician named Blaise Pascal (find out more about Blaise Pascal) – it is helpful when performing Binomial Expansions.

Notice that the 5th row, for example, has 6 entries. The first entry in any one row is known as the 0th entry. Similarly, the top row with a single 1 is known as the 0th row. This is because it is associated with the expansion of $(x+y)^n$ . Now consider the first number 15 in the 6th row, we call this $^{6}C_{2}$ , pronounced ‘6 choose 2’. This can also be written as ${6}\choose{2}$ . In general, we write $^{n}C_r$ or ${n}\choose {r}$ and is calculated as:

$^{n}C_r=\left(\begin{array}{c}n\\r\end{array}\right)=\frac{n!}{(n-r)!r!}$

This comes from summing all the appropriate terms above a given entry and simplifies to a fraction with factorials. $^{n}C_r$ can be thought of as the number of combinations of putting $r$ balls in $n$ buckets. It is also the number of times you get an $x^ry^{n-r}$ term in the expansion of $(x+y)^n$ . Hence, this is why Pascal’s triangle is useful in Binomial Expansion. Note that there is a button on your calculator for working out ${n}\choose{r}$ – you don’t necessarily need to calculate the individual factorials. You might also notice that ${{n}\choose{0}}={{n}\choose{n}}=1$ and ${{n}\choose{1}}={{n}\choose{n-1}}=n$ always.

Binomial Expansion

Suppose now that we wish to expand $(x+y)^n$ , i.e. find the Binomial Expansion. In the simple case where $n$ is a relatively small integer value, we can expand the expression one bracket at a time. This is demonstrated in the first two Basic Binomial Expansion Examples in Section 3. Expanding $(x+y)^n$ by hand for larger n becomes a tedious task. The Edexcel Formula Booklet provides the following formula for binomial expansion:

$(a+b)^n=a^n+\left(\begin{array}{c}n\\1\end{array}\right)a^{n-1}b+\left(\begin{array}{c}n\\2\end{array}\right)a^{n-2}b^2+...+\left(\begin{array}{c}n\\r\end{array}\right)a^{n-r}b^r+...+b^n$

for $n\in {\mathbb N}$ , where:

$\left(\begin{array}{c}n\\r\end{array}\right)=\frac{n!}{(n-r)!r!}$

is $n$ choose $r$ as we saw earlier. Note that this expansion is only true for when $n\in{\mathbb N}$ , i.e for when n is a positive integer. Directly substituting $x$ in place of $a$ and $y$ in place of $b$ results in finding the expansions of $(x+y)^n$ for larger $n$ . Usually, only the first few terms are required. You may substitute other expressions or numbers for $a$ and $b$ but note that when there are added coefficients, the expanded coefficients change dramatically from those given in Pascal’s triangle. Also beware when the question asks you for descending powers of $x$ – you may need to swap the variables accordingly.

Relationship to Binomial Probabilities

Before moving on to the examples section, take a moment to consider the relationship between Binomial Expansion and Binomial Distribution. Consider a binomially distributed random variable with $n$ trials and probability of success $p$ , that is, $X\sim B(n,p)$ . If we require $r$ of the trials to be successful (with probability $p^r$ ) then we also require the remaining $n-r$ trials to be unsuccessful (with probability $(1-p)^{n-r}$ ). The number of combinations in which there can be $r$ successes out of $n$ trials is ${n}\choose{r}$ (see Pascal’s Triangle and ‘n choose r’ notation in the notes section). It follows that the associated probability is given by: